Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / h / Solution

Write $D={\rho }_s/s$, so ${\omega }_k=D{k}^2$, and take ${\Gamma }_k=\gamma |k|$. The small-wavenumber static structure factor behaves asThe spatial Fourier transform of $|k{|}^{-1}$ in $d$ dimensions scales as $|x{|}^{1-d}$. Thusand in three dimensions $C(x)\sim |x{|}^{-2}$. Since this two-point correlation function tends to zero rather than to a nonzero constant at large separation, it is incompatible with true long-range ferromagnetic order and hence with spontaneous symmetry breaking at this higher temperature.